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FAMILIES OF GROUP ACTIONS, GENERIC ISOTRIVIALITY, AND LINEARIZATION

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Abstract

Our base field is the field ℂ of complex numbers. We study families of reductive group actions on \( {\mathbb A} \)2 parametrized by curves and show that every faithful action of a non-finite reductive group on \( {\mathbb A} \)3 is linearizable, i.e., G-isomorphic to a representation of G. The difficulties arise for non-connected groups G.

We prove a Generic Equivalence Theorem which says that two affine morphisms 𝑝: S ⟶ Y and q : Τ ⟶ Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant étale base change φ: U ⟶ Y . A special case is the following result. Call a morphism φ: X ⟶ Y a fibration with fiber F if φ is at and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an étale dominant morphism μ: U ⟶ Y such that the pull-back is a trivial fiber bundle: U × Y X ≅ U × F.

As an application we give short proofs of the following two (known) results:

(a) Every affine A1-_bration over a normal variety is locally trivial in the Zariskitopology (see [KW85]).

(b) Every affine A2-_bration over a smooth curve is locally trivial in the ZariskiTopology (see [KZ01]).

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Authors and Affiliations

  1. Mathematisches Institut, Universität Basel, Rheinsprung 21 CH-4051, Basel, Switzerland

    HANSPETER KRAFT

  2. Department of Mathematics and Statistics McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    PETER RUSSELL

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  1. HANSPETER KRAFT
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  2. PETER RUSSELL
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Correspondence to HANSPETER KRAFT.

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*Supported by SNF (Schweizerischer Nationalfonds).

**Supported by NSERC, Canada.

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KRAFT, H., RUSSELL, P. FAMILIES OF GROUP ACTIONS, GENERIC ISOTRIVIALITY, AND LINEARIZATION. Transformation Groups 19, 779–792 (2014). https://doi.org/10.1007/s00031-014-9274-9

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